enter image description hereI'm reading Stillwell's Elements of Algebra (see screen). I'm struggling with proving the equivalence of the versions of induction, specifically the step (II=>III).
If I take T = {1,2}. T is a subset of N and 1 is its least member. Then looking at the line II=>III. "Let S = N-T" (so S={0,3,4,...}). "If T has no least member then 0 is a member of S" (T has a least member so this part doesn't apply), "otherwise 0 is the least member of T" No! 1 is the least member of T! What am I missing?
The sentence that puzzles you is indeed awkwardly written. A better way to say it might be
The argument then goes on to show that knowing $0 \in S$ leads to a contradiction.